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To Identify and Sketch the graph of a Rational Function with Denominator Degree One
Using Transformations.
INVESTIGATION:
Use technology to investigate the transformations between
f ( ) x x
and transform using f(x) = af(b(x – h) + k
Graph the following using technology:
a)
f ( ) x x
Why does the definition of a rational function specify that q x 0?
b)
f ( ) x x
or this could be written as
f ( ) x 3 x
.
What transformative variable does the 3 represent in this question? ______ What type of transformation is it?
_____________ What is the mapping? (x, y) ( , )
c)
y x
or this could be written as
y x
.
What transformative variable does the 4 represent in this question? _____ What type of transformation is it?
______________ What is the entire mapping? (x, y) ( , )
d)
y x
.
What transformative variable does the 5 represent in this question? ______ What type of transformation is it?
_____________ What is the entire mapping? (x, y) ( , )
PC 30 9.1A Exploring Rational Functions Using Transformations
RATIONAL FUNCTION: A RATIONAL FUNCTION is a function f(x) of the form
p x f x q x
where p(x) and q(x)
are polynomial expressions and q x ( ) 0. The base rational function is written as
f ( ) x x
Given a base RATIONAL FUNCTION of
f ( ) x x
, we can transform it in the same way we transformed
other base functions such as
2 y x , y x , y x
The transformation for RATIONAL FUNCTIONS can be written specifically as a new function
a g x k b x h
Example #1: Sketch
f ( ) x x
using a table of values.
What happens to the graph as the value of x gets
very close (approaches) zero from the left?
What about when it approaches zero from the right?
What happens to the graph as the value of y
approaches zero from above?
What about when it approaches zero from
below?
x y
0
½
1
2
4
BASE FUNCTION CHARACTERISTICS:
f ( ) x x
non-permissible value(s):
behaviour near non-permissible value(s):
end behaviour:
domain:
range:
equation of vertical asymptote:
equation of horizontal asymptote:
x intercept:
y intercept:
An asymptote is a line that the graph of a relation approaches as a limit or boundary.
The graph of a rational function never crosses a vertical asymptote but it may or may not cross a
horizontal asymptote.
Asymptotes are drawn using dashed lines
At what values do vertical asymptotes occur?
Equation of the vertical asymptote of
f x x
Equation of the horizontal asymptote of
f x x
The end behaviour is what happens when x approaches a large positive value or a large negative
value( )
Example #3:
Graph the following function using mappings and identify its characteristics.
x y x
NOTE: The equation is not given in the standard form. Rewrite it in the form y = af(b(x – h)) + k or ( )
( )
a g x k b x h
Describe how to find the vertical asymptote:
How did we find the horizontal asymptote in
this question:
Is there an easier way to find the equation of
the horizontal asymptote if we don’t need to
rewrite the equation into standard form?
x y
0
½
1
2
4
CHARACTERISTIC f(x) =
MAPPING:
Non-Permissible Values
(NPV)
Behavior near NPV
End Behavior
Domain
Range
Vertical Asymptote
Horizontal Asymptote
X Intercept
Y Intercept
Example #3:
a) Sketch a graph of the function
x y x
by finding
the x intercepts
the y intercepts (and another test point)
the vertical and horizontal asymptotes.
b) Can you write the equation in standard form?
Example #4:
Write the equation for the function in the form
a y k b x h
9.1A FA: P442 #1, 3, 4ab(no technology), 5
9.1A MLA: P442 #9, 10, 11
9.1A ULA: 20, 21
Example #2:
a) Sketch the graph of the function 2
y x x
by transforming the graph of 2
y x
and using mapping
b) Manipulate this equation so that it is in the form of a rational function of a single fraction
p x f x q x
x f(x)
0
.
1
2
CHARACTERISTIC f(x) =
MAPPING:
Non-Permissible Values (NPV)
Behavior near NPV
End Behavior
Domain
Range
Vertical Asymptote
Horizontal Asymptote
X Intercept
Y Intercept
Example #3: The rational function
5
a y k x
passes through the points (3, 5) and (7, 2). Determine the
values of a and k.
Example #3: A mobile phone service provider offers several different prepaid plans. One of the plans has a $
monthly fee and a rate of 10¢per text message sent or minute of talk time. Another plan has a monthly fee of $5 and a
rate of 15¢ per text message sent or minute of talk time. Talk time is billed per whole minute.
a) Represent the average cost per text or minute of each plan with a rational function.
b) Graph the functions using technology.
c) What do the graphs show about the average cost per
text or minute for these two plans as the number of
texts and minutes changes?
d) Which plan is the better choice?
9.1B FA: P442 #2, 6, 7a
9.1B ULA: P442 #7bd, 8, 13, 14, 16, 17
9.1B ULA: P442 #19, 22, C
Example #1: Use the equation of each function to predict whether its graph has a horizontal or oblique
asymptote. Find their equations.
a) 2 9
x y x
b)
2 4
x y x
INVESTIGATION:
Consider the function
2 3
x x y x
.
Would you expect it to be a linear graph, a quadratic graph or a graph with asymptotes
(rational graph)?
Can you predict any of the characteristics of the graph?
Use technology to graph the function. Any surprises?
Go to your table function – is there anything interesting that happens at x = 3?
Sketch the graph showing all important details. Use algebra to determine the resulting equation of the original
function.
Example #2: Use the equation of each function to predict whether its graph has vertical asymptote(s), point(s) of
discontinuity or both.
a) 2 9
x y x
b)
2
2
x y x x
c)
3 1
x y x
VERTICAL ASYMPTOTOTES AND POINTS OF DISCONTINUITY (HOLES) FOR RATIONAL
FUNCTIONS: A RATIONAL FUNCTION in the form
p x f x q x
where p(x) and q(x) are polynomial expressions
and q x ( ) 0 and where p(x) and q(x) are BOTH IN FACTORED FORM
A. Vertical Asymptote(s) will exist if there are no common factors between p(x) and q(x). The equation(s) of the
asymptote(s) are identified by the non-permissible values of the function.
EXAMPLE:
2 2
x x y x
B. Points of Discontinuity (Holes in the Graph) will exist if there is at least one common factor between p(x) and q(x).
To find these points you solve the common factors by setting them equal to zero.
C.
EXAMPLE:
2 6
x x y x
D. BOTH VERTICAL ASYMPTOTE(S) AND POINT(S) OF DISCONTINUITY: If you remove common factors and still have
a rational expression (if you still have an x in the denominator), you have both point(s) of discontinuity and vertical
asymptote(s).
EXAMPLE:
2
2
x x y x x
Example #4: Compare the behaviours of
2 2 ( ) 4 2
x x f x x
and
2 2 ( ) 4 2
x x g x x
2 2 ( ) 4 2
x x f x x
2 2 ( ) 4 2
x x g x x
Non Permissible Value(s) (NPV)
What feature exists at the NPV
What is the behaviour at the NPV
Example #5: Match the following. Justify your answers.
Example #6: Find the equations for the following graphs of rational functions. In order to find the equation be
sure to first find the following: Point(s) of Discontinuity, Vertical Asymptote(s) , Horizontal Asymptote, X intercept(s), Y
intercept
a)
b)
9.2 FA: P451 #1, 4abc, 5, 6
9.2 MLA: P451 #4d, 7, 8, 9, 12, 14, 15, 16 (#23 USING TECHNOLOGY)
9.2 ULA: P451 #10, 11, 17, 19, 20, 21
Example #3: Solve
x x x x x
Example #3: Amber and Matteo are travelling separately from their home in Calgary to a wedding 400 km away.
Amber leaves 1 h earlier than Matteo, but Matteo drives at an average speed 20 km/h faster than Amber. If they arrive
at the wedding at the exact same time, what was the average speed at which each of them travelled?
a) Let x represent the time it takes Amber to travel to the wedding. Write an expression for the average speed
that each person travels.
b) Write and solve an equation that represents the difference in their average speeds.
9.3 FA: P465 #3, 5ac(algebraically), 6abd(algebraically)
9.3 MLA: P465 # 4 (using technology), 7, 8, 10, 11, 13,
9.3 ULA: P465 #16*, 17, 15
LIST OF VIDEOS THAT MAY AID IN UNDERSTANDING
Section 9.
https://goo.gl/Tdtbm
https://goo.gl/hMvr8K
Section 9. 2
https://goo.gl/cpYSdm
https://goo.gl/kKdXwr
Section 9.
https://goo.gl/gqL8Kg
